Rank-One Coherence Obstructions in High–High Navier–Stokes Interactions
Internal ID: CGI-RSR-000005
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics, Physics
Research Topics: Partial differential equations / fluid dynamics, Clay Millennium Problems, Navier-Stokes
Abstract
We study comparable high–high interactions in the three-dimensional incompressible Navier–Stokes nonlinearity. For each dyadic output cell, we refine the interaction by angular sector, helicity, radial sub-shell, and output polarization, and form the Gram matrix of the corresponding output contributions. We call this Gram matrix the extended output coherence matrix. Its top eigenvalue measures rank-one coherent output summation, while the remaining spectrum measures secondary independent output channels. The main result is a high–high coherence closure theorem. If a positive fraction of the Gram spectrum lies away from the top eigenvalue, then a spectral off-diagonal bilinear coherence inequality gives a strict gap from rank-one coherent saturation. Near the rank-one Gram locus, the Navier–Stokes symbol forces a structural degeneracy: same-helicity same-radius coherence is Beltrami-depleted through the Lamb-vector identity, mixed-helicity and radially separated interactions split into orthogonal extended channels, and the only remaining finite radially separated beat is parabolically damped unless replenished by additional channels. Consequently, nondepleted comparable high–high Navier–Stokes interactions cannot sustain rank-one coherent output saturation in this extended Gram-matrix framework. The result is intended as a high–high module for later integration with paraproduct estimates, critical continuation criteria, or profile-decomposition methods.
Available Document
Citation:
Petersen, B. L. (2026). Rank-One Coherence Obstructions in High–High Navier–Stokes Interactions. Zenodo. https://doi.org/10.5281/zenodo.19970064
Source Code and Supporting Materials
N/A
Summary and Notes
The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive directional alignment among The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive coherence among interacting modes.
Rather than asking only how large the velocity field becomes, this approach asks how organized the nonlinear output becomes. In particular, it studies whether high-frequency interactions can collapse into a single dominant output channel capable of sustaining critical saturation.
Related Work
CGI-RSR-000006 : Projective Rank-one Closure for Terminal Navier–Stokes Saturation